Mechanical behaviour of engineering materials

Mechanical Behaviour of Engineering Materials

J. Rösler · H. Harders · M. Bäker

Mechanical Behaviour

of Engineering Materials

Metals, Ceramics, Polymers, and Composites

With 320 Figures and 32 Tables

Prof. Dr. Joachim Rösler

TU Braunschweig

Institut für Werkstoffe

Langer Kamp 8

38106 Braunschweig, Germany

[email protected]

Priv.-Doz. Dr. Martin Bäker

TU Braunschweig

Institut für Werkstoffe

Langer Kamp 8

38106 Braunschweig, Germany

[email protected]

Dr.-Ing. Harald Harders

Gartenstraße 28

45468 Mülheim

Germany

[email protected]

German edition published by the Teubner Verlag Wiesbaden, 2006, ISBN 978-3-8351-0008-4

Library of Congress Control Number:

ISBN 978-3-540-73446-8 Springer Berlin Heidelberg New York

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Prof. Dr. rer. nat. Joachim Rösler, born in 1959, studied materials sci￾ence at the University Stuttgart, Germany, from 1979 to 1985. After earning a

Ph. D. at the Max-Planck Institute for Metals Research, Stuttgart, Germany,

and a post-doctoral fellowship at the University of California, Santa Barbara,

usa, he worked at Asea Brown Boveri ag, Switzerland, from 1991 to 1996,

being finally responsible for the material laboratory of abb Power Genera￾tion Ltd., Switzerland. Since 1996, he has been professor for materials science

and director of the Institute for Materials Science at the Technical University

Braunschweig, Germany. His main research interest lies in high-temperature

materials, the mechanical behaviour of materials, and in materials develop￾ment.

Dr.-Ing. Harald Harders, born in 1972, studied mechanical engineering,

with a focus one mechanics and materials, at the Technical University Braun￾schweig, Germany. In 1999, he worked as research scientist at the German

Aerospace Center (dlr). From 1999 to 2004, he worked as research scientist at

the Institute for Materials Science at the Technical University Braunschweig,

finishing with a Ph.D. thesis (2005) on fatigue of metal foams. Since 2004, he has

been working in the field of life time prediction and modelling of superalloys

and coating systems at Siemens Power Generation in Mülheim an der Ruhr,

Germany.

Priv.-Doz. Dr. rer. nat. Martin Bäker, born in 1966, studied physics

at the University Hamburg, Germany, from 1987 to 1993 and finished his

Ph. D. at the II. Institute for Theoretical Physics of the University Hamburg

in 1995, where he also worked as Post-Doc for a year. Since 1996, he has

been working as research scientist at the Institute for Materials Science at

the Technical University Braunschweig, Germany, focusing on continuum me￾chanics simulation of materials. In 2004, he finished his ‘habilitation’ (lecturer

qualification) in the field of materials science.

By the authors

Preface

Components used in mechanical engineering usually have to bear high me￾chanical loads. It is, thus, of considerable importance for students of mechan￾ical engineering and materials science to thoroughly study the mechanical

behaviour of materials. There are different approaches to this subject: The en￾gineer is mainly interested in design rules to dimension components, whereas

materials science usually focuses on the physical processes in the material

occurring during mechanical loading. Ultimately, however, both aspects are

important in practice. Without a clear understanding of the mechanisms of

deformation in the material, the engineer might uncritically apply design rules

and thus cause ‘unexpected’ failure of components. On the other hand, all the￾oretical knowledge is practically useless if the gap to practical application is

not closed.

Our objective in writing this book is to help in solving this problem. For

this reason, the topics covered range from the treatment of the mechanisms

of deformation under mechanical loads to the engineering practice in dimen￾sioning components. To meet the needs of modern engineering, which is more

than ever characterised by the use of all classes of materials, we also needed to

discuss the peculiarities of metals, ceramics, polymers, and composites. This is

reflected in the structure of the book. On the one hand, there are some chap￾ters dealing with the different types of mechanical loading common to several

classes of materials (Chapter 2, elastic behaviour; Chapter 3, plasticity and

failure; Chapter 4, notches; Chapter 5, fracture mechanics; Chapter 10, fa￾tigue; Chapter 11, creep). The specifics of the mechanical behaviour of the

different material classes that are due to their structure and the resulting mi￾crostructural processes are treated in separate chapters (Chapter 6, metals;

Chapter 7, ceramics; Chapter 8, polymers; Chapter 9, composites).

In this book, we thus aim to comprehensively cover the mechanical be￾haviour of materials. It addresses students of mechanical engineering and ma￾terials science as well as practising engineers working on the design of compo￾nents. Although the book contains an in-depth treatment of the mechanical

behaviour and is thus not to be considered as an introduction, all topics can

VIII Preface

be understood without much previous knowledge of material physics and me￾chanics. To make it more accessible, the book starts with an introductory

chapter on the structure of materials and contains appendices on tensors,

crystal orientation, and thermodynamics.

In many cases, we thought it desirable to cover some topics in greater depth

for those readers with a special interest in the subject matter. These sections

can be skipped without compromising the understanding of other subjects.

These advanced sections are indented, as here, or, in the case of longer

sections, marked with a ∗ on the section number.

At the end of the main part, the reader can find some exercises with complete

solutions. They serve as numerical examples for the topics covered in the text

and enable the reader to check their understanding of the subject.

This book has evolved from lectures at the Technical University of

Braunschweig on the mechanical behaviour of materials, aimed at graduate

students, and was first published in German by the Teubner Verlag, Wies￾baden. Due to its success and many encouraging remarks from readers, it

seemed worthwhile to prepare an English edition of the book. In doing so,

the nomenclature and some of the references were adapted to improve the

usability of the book for English readers.

We wish to thank G¨unter Lange who provided valuable help in prepar￾ing this book. Furthermore, we want to thank J¨urgen Huber (CeramTec ag),

Dr. Peter Neumann (Max-Planck-Institut f¨ur Eisenforschung GmbH), Volker

Saß (ThyssenKrupp Nirosta GmbH), Johannes Stoiber (Allianz-Zentrum f¨ur

Technik GmbH), the Lufthansa Technik ag, the Institut f¨ur Werkstofftech￾nik of the Universit¨at Gh Kassel, the Institut f¨ur F¨uge- und Schweißtechnik

of the Technische Universit¨at Braunschweig, the Institut f¨ur Baustoffe, Mas￾sivbau und Brandschutz of the Technische Universit¨at Braunschweig, and all

members of the Institut f¨ur Werkstoffe. Steffen M¨uller has made a signifi￾cant contribution to the lecture notes that were the starting point for writing

this book. Furthermore, we want to thank Allister James and Gary Merrill

who proofread parts of the manuscript. We are also indebted to many read￾ers who sent book evaluations to the Teubner Verlag that have been helpful

in preparing the second German edition [123]. The Teubner Verlag kindly

gave the permission to publish an English translation. We finally want to

thank the Springer publishing company for the cooperation in preparing this

edition.

Braunschweig, Joachim R¨osler

M¨ulheim an der Ruhr, Harald Harders

May 2007 Martin B¨aker

Contents

1 The structure of materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Atomic structure and the chemical bond. . . . . . . . . . . . . . . . . . . . 1

1.2 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Metallic bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Crystal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.3 Polycrystalline metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.1 Covalent bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.2 Ionic bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.3 Dipole bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3.4 Van der Waals bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3.5 Hydrogen bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.6 The crystal structure of ceramics . . . . . . . . . . . . . . . . . . . . 21

1.3.7 Amorphous ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4.1 The chemical structure of polymers . . . . . . . . . . . . . . . . . . 24

1.4.2 The structure of polymers . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.1 Deformation modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Stress and strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.2 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Atomic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4 Hooke’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4.1 Elastic strain energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

∗ 2.4.2 Elastic deformation under multiaxial loads1

. . . . . . . . . . . 43

∗ 2.4.3 Isotropic material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

1 Sections with a title marked by a ∗ contain advanced information which can be

skipped without impairing the understanding of subsequent topics.

X Contents

∗ 2.4.4 Cubic lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

∗ 2.4.5 Orthorhombic crystals and orthotropic elasticity. . . . . . . 53

∗ 2.4.6 Transversally isotropic elasticity . . . . . . . . . . . . . . . . . . . . . 54

∗ 2.4.7 Other crystal lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

∗ 2.4.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

∗ 2.5 Isotropy and anisotropy of macroscopic components . . . . . . . . . . 57

2.6 Temperature dependence of Young’s modulus . . . . . . . . . . . . . . . 60

3 Plasticity and failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.1 Nominal and true strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2 Stress-strain diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.1 Types of stress-strain diagrams . . . . . . . . . . . . . . . . . . . . . . 68

3.2.2 Analysis of a stress-strain diagram . . . . . . . . . . . . . . . . . . . 73

3.2.3 Approximation of the stress-strain curve. . . . . . . . . . . . . . 81

3.3 Plasticity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.3.1 Yield criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.3.2 Yield criteria of metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.3.3 Yield criteria of polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.3.4 Flow rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.3.5 Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

∗ 3.3.6 Application of a yield criterion, flow rule, and

hardening rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

∗ 3.4 Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

∗ 3.4.1 Scratch tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

∗ 3.4.2 Indentation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

∗ 3.4.3 Rebound tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.5 Material failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.5.1 Shear fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.5.2 Cleavage fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.5.3 Fracture criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4 Notches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.1 Stress concentration factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.2 Neuber’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

∗ 4.3 Tensile testing of notched specimens . . . . . . . . . . . . . . . . . . . . . . . 125

5 Fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.1 Introduction to fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.2 Linear-elastic fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.2.1 The stress field near a crack tip . . . . . . . . . . . . . . . . . . . . . 131

5.2.2 The energy balance of crack propagation . . . . . . . . . . . . . 134

5.2.3 Dimensioning pre-cracked components

under static loads 142

5.2.4 Fracture parameters of different materials . . . . . . . . . . . . 144

5.2.5 Material behaviour during crack propagation. . . . . . . . . . 146

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents XI

∗ 5.2.6 Subcritical crack propagation . . . . . . . . . . . . . . . . . . . . . . . 150

∗ 5.2.7 Measuring fracture parameters . . . . . . . . . . . . . . . . . . . . . . 152

∗ 5.3 Elastic-plastic fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 158

∗ 5.3.1 Crack tip opening displacement (ctod) . . . . . . . . . . . . . . 158

∗ 5.3.2 J integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

∗ 5.3.3 Material behaviour during crack propagation. . . . . . . . . . 161

∗ 5.3.4 Measuring elastic-plastic fracture mechanics parameters 163

6 Mechanical behaviour of metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.1 Theoretical strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.2 Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.2.1 Types of dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.2.2 The stress field of a dislocation . . . . . . . . . . . . . . . . . . . . . . 168

6.2.3 Dislocation movement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.2.4 Slip systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.2.5 The critical resolved shear stress . . . . . . . . . . . . . . . . . . . . 178

6.2.6 Taylor factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.2.7 Dislocation interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.2.8 Generation, multiplication and annihilation of

dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6.2.9 Forces acting on dislocations . . . . . . . . . . . . . . . . . . . . . . . . 187

6.3 Overcoming obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.3.1 Athermal processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

6.3.2 Thermally activated processes . . . . . . . . . . . . . . . . . . . . . . . 193

6.3.3 Ductile-brittle transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

6.3.4 Climb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

6.3.5 Intersection of dislocations. . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.4 Strengthening mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

6.4.1 Work hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

6.4.2 Grain boundary strengthening . . . . . . . . . . . . . . . . . . . . . . 200

6.4.3 Solid solution hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

6.4.4 Particle strengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

6.4.5 Hardening of steels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

∗ 6.5 Mechanical twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

7 Mechanical behaviour of ceramics . . . . . . . . . . . . . . . . . . . . . . . . . 227

7.1 Manufacturing ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

7.2 Mechanisms of crack propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 229

7.2.1 Crack deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

7.2.2 Crack bridging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

7.2.3 Microcrack formation and crack branching . . . . . . . . . . . . 231

7.2.4 Stress-induced phase transformations . . . . . . . . . . . . . . . . 232

7.2.5 Stable crack growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

∗ 7.2.6 Subcritical crack growth in ceramics . . . . . . . . . . . . . . . . . 234

7.3 Statistical fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

XII Contents

7.3.1 Weibull statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

∗ 7.3.2 Weibull statistics for subcritical crack growth . . . . . . . . . 242

∗ 7.3.3 Measuring the parameters σ0 and m . . . . . . . . . . . . . . . . . 243

∗ 7.4 Proof test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

7.5 Strengthening ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

7.5.1 Reducing defect size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

7.5.2 Crack deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

7.5.3 Microcracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

7.5.4 Transformation toughening . . . . . . . . . . . . . . . . . . . . . . . . . 252

7.5.5 Adding ductile particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

8 Mechanical behaviour of polymers . . . . . . . . . . . . . . . . . . . . . . . . . 257

8.1 Physical properties of polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

8.1.1 Relaxation processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

8.1.2 Glass transition temperature . . . . . . . . . . . . . . . . . . . . . . . . 260

8.1.3 Melting temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

8.2 Time-dependent deformation of polymers . . . . . . . . . . . . . . . . . . . 263

8.2.1 Phenomenological description of time-dependence . . . . . 263

8.2.2 Time-dependence and thermal activation . . . . . . . . . . . . . 266

8.3 Elastic properties of polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

8.3.1 Elastic properties of thermoplastics . . . . . . . . . . . . . . . . . . 269

8.3.2 Elastic properties of elastomers and duromers . . . . . . . . . 273

8.4 Plastic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

8.4.1 Amorphous thermoplastics . . . . . . . . . . . . . . . . . . . . . . . . . 275

8.4.2 Semi-crystalline thermoplastics . . . . . . . . . . . . . . . . . . . . . . 281

8.5 Increasing the thermal stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

8.5.1 Increasing the glass and the melting temperature . . . . . . 284

8.5.2 Increasing the crystallinity. . . . . . . . . . . . . . . . . . . . . . . . . . 287

8.6 Increasing strength and stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

8.7 Increasing the ductility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

∗ 8.8 Environmental effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

9 Mechanical behaviour of fibre reinforced composites . . . . . . . 295

9.1 Strengthening methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

9.1.1 Classifying by particle geometry . . . . . . . . . . . . . . . . . . . . . 296

9.1.2 Classifying by matrix systems . . . . . . . . . . . . . . . . . . . . . . . 299

9.2 Elasticity of fibre composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

9.2.1 Loading in parallel to the fibres . . . . . . . . . . . . . . . . . . . . . 301

9.2.2 Loading perpendicular to the fibres . . . . . . . . . . . . . . . . . . 301

∗ 9.2.3 The anisotropy in general . . . . . . . . . . . . . . . . . . . . . . . . . . 302

9.3 Plasticity and fracture of composites . . . . . . . . . . . . . . . . . . . . . . . 303

9.3.1 Tensile loading with continuous fibres . . . . . . . . . . . . . . . . 303

9.3.2 Load transfer between matrix and fibre . . . . . . . . . . . . . . 305

9.3.3 Crack propagation in fibre composites . . . . . . . . . . . . . . . . 308

9.3.4 Statistics of composite failure . . . . . . . . . . . . . . . . . . . . . . . 312

Contents XIII

9.3.5 Failure under compressive loads . . . . . . . . . . . . . . . . . . . . . 313

9.3.6 Matrix-dominated failure and arbitrary loads . . . . . . . . . 315

9.4 Examples of composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

9.4.1 Polymer matrix composites . . . . . . . . . . . . . . . . . . . . . . . . . 315

9.4.2 Metal matrix composites . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

9.4.3 Ceramic matrix composites . . . . . . . . . . . . . . . . . . . . . . . . . 323

∗ 9.4.4 Biological composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

10 Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

10.1 Types of loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

10.2 Fatigue failure of metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

10.2.1 Crack initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

10.2.2 Crack propagation (stage II) . . . . . . . . . . . . . . . . . . . . . . . . 342

10.2.3 Final fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

10.3 Fatigue of ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

10.4 Fatigue of polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

10.4.1 Thermal fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

10.4.2 Mechanical fatigue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

10.5 Fatigue of fibre composites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

10.6 Phenomenological description of the fatigue strength . . . . . . . . . 349

10.6.1 Fatigue crack growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

10.6.2 Stress-cycle diagrams (S-N diagrams) . . . . . . . . . . . . . . . . 357

10.6.3 The role of mean stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

∗ 10.6.4 Fatigue assessment with variable amplitude loading . . . . 368

∗ 10.6.5 Cyclic stress-strain behaviour . . . . . . . . . . . . . . . . . . . . . . . 369

∗ 10.6.6 Kitagawa diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

∗ 10.7 Fatigue of notched specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

11 Creep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

11.1 Phenomenology of creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

11.2 Creep mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

11.2.1 Stages of creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

11.2.2 Dislocation creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

11.2.3 Diffusion creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

11.2.4 Grain boundary sliding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

11.2.5 Deformation mechanism maps . . . . . . . . . . . . . . . . . . . . . . 396

11.3 Creep fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

11.4 Increasing the creep resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

1 Packing density of crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

2 Macromolecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

3 Interaction between two atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

4 Bulk modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

5 Relation between the elastic constants . . . . . . . . . . . . . . . . . . . . . 408

XIV Contents

6 Candy catapult . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

7 True strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

8 Interest calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

9 Large deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

10 Yield criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

11 Yield criteria of polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

12 Design of a notched shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

13 Estimating the fracture toughness KIc . . . . . . . . . . . . . . . . . . . . . 412

14 Determination of the fracture toughness KIc . . . . . . . . . . . . . . . . 412

15 Static design of a tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

16 Theoretical strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

17 Estimating the dislocation density . . . . . . . . . . . . . . . . . . . . . . . . . 414

18 Thermally activated dislocation generation . . . . . . . . . . . . . . . . . 414

19 Work hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

20 Grain boundary strengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

21 Precipitation hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

22 Weibull statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

23 Design of a fluid tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

24 Subcritical crack growth of a ceramic component . . . . . . . . . . . 417

25 Mechanical models of viscoelastic polymers . . . . . . . . . . . . . . . . . 417

26 Elastic damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

27 Eyring plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

28 Elasticity of fibre composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

29 Properties of a polymer matrix composite . . . . . . . . . . . . . . . . . . 419

30 Estimating the number of cycles to failure . . . . . . . . . . . . . . . . . . 419

31 Miner’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

32 Larson-Miller parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

33 Creep deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

34 Relaxation of thermal stresses by creep . . . . . . . . . . . . . . . . . . . . . 421

13 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

A Using tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

A.2 The order of a tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

A.3 Tensor notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

A.4 Tensor operations and Einstein summation convention . . . . . . . 453

A.5 Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

A.6 Important constants and tensor operations. . . . . . . . . . . . . . . . . . 457

A.7 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

A.8 Derivations of tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

B Miller and Miller-Bravais indices . . . . . . . . . . . . . . . . . . . . . . . . . . 461

B.1 Miller indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

B.2 Miller-Bravais indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

Contents XV

C A crash course in thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 465

C.1 Thermal activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

C.2 Free energy and free enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

C.3 Phase transformations and phase diagrams . . . . . . . . . . . . . . . . . 468

D The J integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

D.1 Discontinuities, singularities, and Gauss’ theorem . . . . . . . . . . . . 473

D.2 Energy-momentum tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

D.3 J integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

D.4 J integral at a crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

D.5 Plasticity at the crack tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

D.6 Energy interpretation of the J integral . . . . . . . . . . . . . . . . . . . . . 482

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

1

The structure of materials

There is a vast multitude of materials with strongly differing properties. A

copper wire, for instance, can be bent easily into a new shape, whereas a

rubber band will snap back to its initial form after deformation, while the

attempt to bend a glass tube ends with fracture of the tube. The strongly

differing properties are reflected in the application of engineering materials –

you would neither want to build cars of glass nor rubber bridges. The mul￾titude of materials enables the engineer to select the best-suited one for any

particular component. For this, however, it is frequently necessary not only to

know the mechanical properties of the materials, but also to understand the

physical phenomena causing them.

The mechanical properties of materials are determined by their atomic

structure. To understand these properties, some knowledge of the structure of

materials is therefore required. This is the topic covered in this chapter. The

structure of materials is investigated by solid state physics, but to understand

the mechanical properties, it is not necessary to understand the more arcane

aspects of this discipline as they can usually be explained with rather simple

models.

This chapter starts with a short explanation of the basic principles of

atomic structure and the nature of the chemical bond. Afterwards, the three

main groups of materials, metals, ceramics, and polymers, are discussed. The

most important characteristics of their interatomic bonds are covered, and

the microscopic structure of the different groups is also treated.

For a more thorough introduction into the structure of materials the books

by Beiser [17] and Podesta [110] are recommended.

1.1 Atomic structure and the chemical bond

Atoms consist of a positively charged nucleus surrounded by negatively

charged electrons. Almost the complete mass of the atom is concentrated

in the nucleus because it comprises heavy elementary particles, the protons